LMFDB - Embedded newform 4368.2.h.t.337.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4368,2,Mod(337,4368)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 1]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4368.337");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4368.h (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$34.8786556029$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 4x^{9} + 8x^{8} + 18x^{7} + 28x^{6} - 48x^{5} + 130x^{4} + 316x^{3} + 324x^{2} + 144x + 32 $ x^10 - 4x^9 + 8x^8 + 18x^7 + 28x^6 - 48x^5 + 130x^4 + 316x^3 + 324x^2 + 144*x + 32
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 2184)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			337.9
Root			$-0.340491 - 0.340491i$ of defining polynomial
Character	$\chi$	$=$	4368.337
Dual form			4368.2.h.t.337.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{3} +3.19289i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})$	$q+1.00000 q^{3} +3.19289i q^{5} +1.00000i q^{7} +1.00000 q^{9} +5.06675i q^{11} +(-1.56649 + 3.24748i) q^{13} +3.19289i q^{15} +0.320653 q^{17} +0.621083i q^{19} +1.00000i q^{21} +4.19452 q^{23} -5.19452 q^{25} +1.00000 q^{27} -2.30043 q^{29} -0.740881i q^{31} +5.06675i q^{33} -3.19289 q^{35} -7.42708i q^{37} +(-1.56649 + 3.24748i) q^{39} +9.49659i q^{41} +11.0069 q^{43} +3.19289i q^{45} +4.98141i q^{47} -1.00000 q^{49} +0.320653 q^{51} -2.94174 q^{53} -16.1776 q^{55} +0.621083i q^{57} -0.680982i q^{59} -3.13299 q^{61} +1.00000i q^{63} +(-10.3688 - 5.00163i) q^{65} +13.3106i q^{67} +4.19452 q^{69} -13.0668i q^{71} -14.2629i q^{73} -5.19452 q^{75} -5.06675 q^{77} -3.55649 q^{79} +1.00000 q^{81} +9.30207i q^{83} +1.02381i q^{85} -2.30043 q^{87} -13.6470i q^{89} +(-3.24748 - 1.56649i) q^{91} -0.740881i q^{93} -1.98305 q^{95} -4.73026i q^{97} +5.06675i q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 10 q^{3} + 10 q^{9}+O(q^{10}) $ 10 * q + 10 * q^3 + 10 * q^9	$ 10 q + 10 q^{3} + 10 q^{9} + 2 q^{13} - 2 q^{23} - 8 q^{25} + 10 q^{27} - 2 q^{29} - 6 q^{35} + 2 q^{39} + 34 q^{43} - 10 q^{49} - 2 q^{53} - 64 q^{55} + 4 q^{61} + 2 q^{65} - 2 q^{69} - 8 q^{75} + 16 q^{77} + 38 q^{79} + 10 q^{81} - 2 q^{87} + 34 q^{95}+O(q^{100}) $ 10 * q + 10 * q^3 + 10 * q^9 + 2 * q^13 - 2 * q^23 - 8 * q^25 + 10 * q^27 - 2 * q^29 - 6 * q^35 + 2 * q^39 + 34 * q^43 - 10 * q^49 - 2 * q^53 - 64 * q^55 + 4 * q^61 + 2 * q^65 - 2 * q^69 - 8 * q^75 + 16 * q^77 + 38 * q^79 + 10 * q^81 - 2 * q^87 + 34 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4368\mathbb{Z}\right)^\times$.

$n$	$1093$	$1249$	$1457$	$2017$	$3823$
$\chi(n)$	$1$	$1$	$1$	$-1$	$1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	1.00000			0.577350
$3$	1.00000			0.577350
$4$		0			0
$5$			3.19289i			1.42790i	0.700196	+	0.713951i	$0.253096\pi$
$5$			3.19289i			1.42790i	−0.700196	+	0.713951i	$0.746904\pi$
$6$		0			0
$7$			1.00000i			0.377964i
$7$			1.00000i			0.377964i
$8$		0			0
$9$	1.00000			0.333333
$10$		0			0
$11$			5.06675i			1.52768i	0.645403	+	0.763842i	$0.276689\pi$
$11$			5.06675i			1.52768i	−0.645403	+	0.763842i	$0.723311\pi$
$12$		0			0
$13$	−1.56649	+	3.24748i	−0.434467	+	0.900688i
$13$	−1.56649	+	3.24748i	−0.434467	+	0.900688i
$14$		0			0
$15$			3.19289i			0.824400i
$16$		0			0
$17$	0.320653			0.0777697			0.0388849	−	0.999244i	$-0.487619\pi$
$17$	0.320653			0.0777697			0.0388849	+	0.999244i	$0.487619\pi$
$18$		0			0
$19$			0.621083i			0.142486i	0.997459	+	0.0712431i	$0.0226966\pi$
$19$			0.621083i			0.142486i	−0.997459	+	0.0712431i	$0.977303\pi$
$20$		0			0
$21$			1.00000i			0.218218i
$22$		0			0
$23$	4.19452			0.874618			0.437309	−	0.899311i	$-0.355932\pi$
$23$	4.19452			0.874618			0.437309	+	0.899311i	$0.355932\pi$
$24$		0			0
$25$	−5.19452			−1.03890
$26$		0			0
$27$	1.00000			0.192450
$28$		0			0
$29$	−2.30043			−0.427179			−0.213590	−	0.976923i	$-0.568516\pi$
$29$	−2.30043			−0.427179			−0.213590	+	0.976923i	$0.568516\pi$
$30$		0			0
$31$		−	0.740881i		−	0.133066i	−0.997784	−	0.0665331i	$-0.978806\pi$
$31$		−	0.740881i		−	0.133066i	0.997784	−	0.0665331i	$-0.0211938\pi$
$32$		0			0
$33$			5.06675i			0.882009i
$34$		0			0
$35$	−3.19289			−0.539696
$36$		0			0
$37$		−	7.42708i		−	1.22101i	−0.792014	−	0.610503i	$-0.790968\pi$
$37$		−	7.42708i		−	1.22101i	0.792014	−	0.610503i	$-0.209032\pi$
$38$		0			0
$39$	−1.56649	+	3.24748i	−0.250840	+	0.520012i
$40$		0			0
$41$			9.49659i			1.48312i	0.670888	+	0.741559i	$0.265913\pi$
$41$			9.49659i			1.48312i	−0.670888	+	0.741559i	$0.734087\pi$
$42$		0			0
$43$	11.0069			1.67853			0.839265	−	0.543723i	$-0.182986\pi$
$43$	11.0069			1.67853			0.839265	+	0.543723i	$0.182986\pi$
$44$		0			0
$45$			3.19289i			0.475967i
$46$		0			0
$47$			4.98141i			0.726614i	0.931670	+	0.363307i	$0.118352\pi$
$47$			4.98141i			0.726614i	−0.931670	+	0.363307i	$0.881648\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$	0.320653			0.0449004
$52$		0			0
$53$	−2.94174			−0.404078			−0.202039	−	0.979377i	$-0.564757\pi$
$53$	−2.94174			−0.404078			−0.202039	+	0.979377i	$0.564757\pi$
$54$		0			0
$55$	−16.1776			−2.18138
$56$		0			0
$57$			0.621083i			0.0822645i
$58$		0			0
$59$		−	0.680982i		−	0.0886563i	−0.999017	−	0.0443282i	$-0.985885\pi$
$59$		−	0.680982i		−	0.0886563i	0.999017	−	0.0443282i	$-0.0141147\pi$
$60$		0			0
$61$	−3.13299			−0.401138			−0.200569	−	0.979680i	$-0.564279\pi$
$61$	−3.13299			−0.401138			−0.200569	+	0.979680i	$0.564279\pi$
$62$		0			0
$63$			1.00000i			0.125988i
$64$		0			0
$65$	−10.3688	−	5.00163i	−1.28609	−	0.620376i
$66$		0			0
$67$			13.3106i			1.62614i	0.582164	+	0.813072i	$0.302206\pi$
$67$			13.3106i			1.62614i	−0.582164	+	0.813072i	$0.697794\pi$
$68$		0			0
$69$	4.19452			0.504961
$70$		0			0
$71$		−	13.0668i		−	1.55074i	−0.631508	−	0.775369i	$-0.717564\pi$
$71$		−	13.0668i		−	1.55074i	0.631508	−	0.775369i	$-0.282436\pi$
$72$		0			0
$73$		−	14.2629i		−	1.66935i	−0.550745	−	0.834674i	$-0.685656\pi$
$73$		−	14.2629i		−	1.66935i	0.550745	−	0.834674i	$-0.314344\pi$
$74$		0			0
$75$	−5.19452			−0.599812
$76$		0			0
$77$	−5.06675			−0.577410
$78$		0			0
$79$	−3.55649			−0.400136			−0.200068	−	0.979782i	$-0.564116\pi$
$79$	−3.55649			−0.400136			−0.200068	+	0.979782i	$0.564116\pi$
$80$		0			0
$81$	1.00000			0.111111
$82$		0			0
$83$			9.30207i			1.02103i	0.859868	+	0.510517i	$0.170546\pi$
$83$			9.30207i			1.02103i	−0.859868	+	0.510517i	$0.829454\pi$
$84$		0			0
$85$			1.02381i			0.111048i
$86$		0			0
$87$	−2.30043			−0.246632
$88$		0			0
$89$		−	13.6470i		−	1.44658i	−0.690542	−	0.723292i	$-0.742628\pi$
$89$		−	13.6470i		−	1.44658i	0.690542	−	0.723292i	$-0.257372\pi$
$90$		0			0
$91$	−3.24748	−	1.56649i	−0.340428	−	0.164213i
$92$		0			0
$93$		−	0.740881i		−	0.0768258i
$94$		0			0
$95$	−1.98305			−0.203456
$96$		0			0
$97$		−	4.73026i		−	0.480285i	−0.970738	−	0.240143i	$-0.922806\pi$
$97$		−	4.73026i		−	0.480285i	0.970738	−	0.240143i	$-0.0771942\pi$
$98$		0			0
$99$			5.06675i			0.509228i
$100$		0			0
$101$	2.65554			0.264236			0.132118	−	0.991234i	$-0.457822\pi$
$101$	2.65554			0.264236			0.132118	+	0.991234i	$0.457822\pi$
$102$		0			0
$103$	−11.9767			−1.18010			−0.590050	−	0.807366i	$-0.700892\pi$
$103$	−11.9767			−1.18010			−0.590050	+	0.807366i	$0.700892\pi$
$104$		0			0
$105$	−3.19289			−0.311594
$106$		0			0
$107$	4.49495			0.434543			0.217272	−	0.976111i	$-0.430284\pi$
$107$	4.49495			0.434543			0.217272	+	0.976111i	$0.430284\pi$
$108$		0			0
$109$		−	17.3106i		−	1.65805i	−0.559211	−	0.829025i	$-0.688896\pi$
$109$		−	17.3106i		−	1.65805i	0.559211	−	0.829025i	$-0.311104\pi$
$110$		0			0
$111$		−	7.42708i		−	0.704948i
$112$		0			0
$113$	−1.25912			−0.118448			−0.0592240	−	0.998245i	$-0.518863\pi$
$113$	−1.25912			−0.118448			−0.0592240	+	0.998245i	$0.518863\pi$
$114$		0			0
$115$			13.3926i			1.24887i
$116$		0			0
$117$	−1.56649	+	3.24748i	−0.144822	+	0.300229i
$118$		0			0
$119$			0.320653i			0.0293942i
$120$		0			0
$121$	−14.6720			−1.33382
$122$		0			0
$123$			9.49659i			0.856278i
$124$		0			0
$125$		−	0.621083i		−	0.0555514i
$126$		0			0
$127$	−13.3212			−1.18206			−0.591032	−	0.806648i	$-0.701279\pi$
$127$	−13.3212			−1.18206			−0.591032	+	0.806648i	$0.701279\pi$
$128$		0			0
$129$	11.0069			0.969100
$130$		0			0
$131$	16.3890			1.43192			0.715959	−	0.698143i	$-0.245990\pi$
$131$	16.3890			1.43192			0.715959	+	0.698143i	$0.245990\pi$
$132$		0			0
$133$	−0.621083			−0.0538547
$134$		0			0
$135$			3.19289i			0.274800i
$136$		0			0
$137$			5.32229i			0.454714i	0.973812	+	0.227357i	$0.0730084\pi$
$137$			5.32229i			0.454714i	−0.973812	+	0.227357i	$0.926992\pi$
$138$		0			0
$139$	1.45364			0.123296			0.0616480	−	0.998098i	$-0.480364\pi$
$139$	1.45364			0.123296			0.0616480	+	0.998098i	$0.480364\pi$
$140$		0			0
$141$			4.98141i			0.419511i
$142$		0			0
$143$	−16.4542	−	7.93704i	−1.37597	−	0.663728i
$144$		0			0
$145$		−	7.34501i		−	0.609970i
$146$		0			0
$147$	−1.00000			−0.0824786
$148$		0			0
$149$			12.0906i			0.990497i	0.868751	+	0.495249i	$0.164923\pi$
$149$			12.0906i			0.990497i	−0.868751	+	0.495249i	$0.835077\pi$
$150$		0			0
$151$			9.62467i			0.783244i	0.920126	+	0.391622i	$0.128086\pi$
$151$			9.62467i			0.783244i	−0.920126	+	0.391622i	$0.871914\pi$
$152$		0			0
$153$	0.320653			0.0259232
$154$		0			0
$155$	2.36555			0.190006
$156$		0			0
$157$	−14.5495			−1.16117			−0.580586	−	0.814199i	$-0.697177\pi$
$157$	−14.5495			−1.16117			−0.580586	+	0.814199i	$0.697177\pi$
$158$		0			0
$159$	−2.94174			−0.233295
$160$		0			0
$161$			4.19452i			0.330575i
$162$		0			0
$163$			11.0446i			0.865078i	0.901615	+	0.432539i	$0.142382\pi$
$163$			11.0446i			0.865078i	−0.901615	+	0.432539i	$0.857618\pi$
$164$		0			0
$165$	−16.1776			−1.25942
$166$		0			0
$167$			15.4181i			1.19309i	0.802581	+	0.596543i	$0.203459\pi$
$167$			15.4181i			1.19309i	−0.802581	+	0.596543i	$0.796541\pi$
$168$		0			0
$169$	−8.09220	−	10.1743i	−0.622477	−	0.782638i
$170$		0			0
$171$			0.621083i			0.0474954i
$172$		0			0
$173$	−3.29685			−0.250654			−0.125327	−	0.992115i	$-0.539998\pi$
$173$	−3.29685			−0.250654			−0.125327	+	0.992115i	$0.539998\pi$
$174$		0			0
$175$		−	5.19452i		−	0.392669i
$176$		0			0
$177$		−	0.680982i		−	0.0511858i
$178$		0			0
$179$	5.48810			0.410200			0.205100	−	0.978741i	$-0.434248\pi$
$179$	5.48810			0.410200			0.205100	+	0.978741i	$0.434248\pi$
$180$		0			0
$181$	−9.30729			−0.691805			−0.345903	−	0.938270i	$-0.612427\pi$
$181$	−9.30729			−0.691805			−0.345903	+	0.938270i	$0.612427\pi$
$182$		0			0
$183$	−3.13299			−0.231597
$184$		0			0
$185$	23.7138			1.74348
$186$		0			0
$187$			1.62467i			0.118808i
$188$		0			0
$189$			1.00000i			0.0727393i
$190$		0			0
$191$	13.0656			0.945397			0.472698	−	0.881224i	$-0.343280\pi$
$191$	13.0656			0.945397			0.472698	+	0.881224i	$0.343280\pi$
$192$		0			0
$193$		−	15.3757i		−	1.10677i	−0.832927	−	0.553383i	$-0.813337\pi$
$193$		−	15.3757i		−	1.10677i	0.832927	−	0.553383i	$-0.186663\pi$
$194$		0			0
$195$	−10.3688	−	5.00163i	−0.742527	−	0.358175i
$196$		0			0
$197$		−	10.8961i		−	0.776313i	−0.921593	−	0.388156i	$-0.873112\pi$
$197$		−	10.8961i		−	0.776313i	0.921593	−	0.388156i	$-0.126888\pi$
$198$		0			0
$199$	17.5945			1.24724			0.623620	−	0.781727i	$-0.285661\pi$
$199$	17.5945			1.24724			0.623620	+	0.781727i	$0.285661\pi$
$200$		0			0
$201$			13.3106i			0.938854i
$202$		0			0
$203$		−	2.30043i		−	0.161459i
$204$		0			0
$205$	−30.3215			−2.11775
$206$		0			0
$207$	4.19452			0.291539
$208$		0			0
$209$	−3.14688			−0.217674
$210$		0			0
$211$	20.9354			1.44125			0.720627	−	0.693323i	$-0.243854\pi$
$211$	20.9354			1.44125			0.720627	+	0.693323i	$0.243854\pi$
$212$		0			0
$213$		−	13.0668i		−	0.895319i
$214$		0			0
$215$			35.1436i			2.39678i
$216$		0			0
$217$	0.740881			0.0502943
$218$		0			0
$219$		−	14.2629i		−	0.963798i
$220$		0			0
$221$	−0.502300	+	1.04131i	−0.0337884	+	0.0700462i
$222$		0			0
$223$		−	19.1267i		−	1.28082i	−0.768035	−	0.640408i	$-0.778765\pi$
$223$		−	19.1267i		−	1.28082i	0.768035	−	0.640408i	$-0.221235\pi$
$224$		0			0
$225$	−5.19452			−0.346301
$226$		0			0
$227$		−	29.7218i		−	1.97270i	−0.164653	−	0.986352i	$-0.552650\pi$
$227$		−	29.7218i		−	1.97270i	0.164653	−	0.986352i	$-0.447350\pi$
$228$		0			0
$229$			9.99265i			0.660333i	0.943923	+	0.330166i	$0.107105\pi$
$229$			9.99265i			0.660333i	−0.943923	+	0.330166i	$0.892895\pi$
$230$		0			0
$231$	−5.06675			−0.333368
$232$		0			0
$233$	24.9253			1.63291			0.816455	−	0.577410i	$-0.195936\pi$
$233$	24.9253			1.63291			0.816455	+	0.577410i	$0.195936\pi$
$234$		0			0
$235$	−15.9051			−1.03753
$236$		0			0
$237$	−3.55649			−0.231019
$238$		0			0
$239$		−	9.66762i		−	0.625346i	−0.949861	−	0.312673i	$-0.898776\pi$
$239$		−	9.66762i		−	0.625346i	0.949861	−	0.312673i	$-0.101224\pi$
$240$		0			0
$241$			23.2867i			1.50003i	0.661421	+	0.750015i	$0.269954\pi$
$241$			23.2867i			1.50003i	−0.661421	+	0.750015i	$0.730046\pi$
$242$		0			0
$243$	1.00000			0.0641500
$244$		0			0
$245$		−	3.19289i		−	0.203986i
$246$		0			0
$247$	−2.01695	−	0.972923i	−0.128336	−	0.0619056i
$248$		0			0
$249$			9.30207i			0.589494i
$250$		0			0
$251$	−4.98990			−0.314960			−0.157480	−	0.987522i	$-0.550337\pi$
$251$	−4.98990			−0.314960			−0.157480	+	0.987522i	$0.550337\pi$
$252$		0			0
$253$			21.2526i			1.33614i
$254$		0			0
$255$			1.02381i			0.0641133i
$256$		0			0
$257$	−18.7923			−1.17223			−0.586116	−	0.810227i	$-0.699344\pi$
$257$	−18.7923			−1.17223			−0.586116	+	0.810227i	$0.699344\pi$
$258$		0			0
$259$	7.42708			0.461497
$260$		0			0
$261$	−2.30043			−0.142393
$262$		0			0
$263$	−10.3110			−0.635807			−0.317903	−	0.948123i	$-0.602979\pi$
$263$	−10.3110			−0.635807			−0.317903	+	0.948123i	$0.602979\pi$
$264$		0			0
$265$		−	9.39263i		−	0.576984i
$266$		0			0
$267$		−	13.6470i		−	0.835186i
$268$		0			0
$269$	10.0821			0.614716			0.307358	−	0.951594i	$-0.400555\pi$
$269$	10.0821			0.614716			0.307358	+	0.951594i	$0.400555\pi$
$270$		0			0
$271$			20.9105i			1.27023i	0.772419	+	0.635113i	$0.219046\pi$
$271$			20.9105i			1.27023i	−0.772419	+	0.635113i	$0.780954\pi$
$272$		0			0
$273$	−3.24748	−	1.56649i	−0.196546	−	0.0948085i
$274$		0			0
$275$		−	26.3194i		−	1.58712i
$276$		0			0
$277$	11.8610			0.712660			0.356330	−	0.934360i	$-0.384028\pi$
$277$	11.8610			0.712660			0.356330	+	0.934360i	$0.384028\pi$
$278$		0			0
$279$		−	0.740881i		−	0.0443554i
$280$		0			0
$281$			18.0534i			1.07697i	0.842634	+	0.538487i	$0.181004\pi$
$281$			18.0534i			1.07697i	−0.842634	+	0.538487i	$0.818996\pi$
$282$		0			0
$283$	−6.50557			−0.386716			−0.193358	−	0.981128i	$-0.561938\pi$
$283$	−6.50557			−0.386716			−0.193358	+	0.981128i	$0.561938\pi$
$284$		0			0
$285$	−1.98305			−0.117466
$286$		0			0
$287$	−9.49659			−0.560566
$288$		0			0
$289$	−16.8972			−0.993952
$290$		0			0
$291$		−	4.73026i		−	0.277293i
$292$		0			0
$293$		−	22.7392i		−	1.32844i	−0.747537	−	0.664220i	$-0.768764\pi$
$293$		−	22.7392i		−	1.32844i	0.747537	−	0.664220i	$-0.231236\pi$
$294$		0			0
$295$	2.17430			0.126593
$296$		0			0
$297$			5.06675i			0.294003i
$298$		0			0
$299$	−6.57069	+	13.6216i	−0.379993	+	0.787758i
$300$		0			0
$301$			11.0069i			0.634425i
$302$		0			0
$303$	2.65554			0.152557
$304$		0			0
$305$		−	10.0033i		−	0.572786i
$306$		0			0
$307$			31.3959i			1.79186i	0.444197	+	0.895929i	$0.353489\pi$
$307$			31.3959i			1.79186i	−0.444197	+	0.895929i	$0.646511\pi$
$308$		0			0
$309$	−11.9767			−0.681331
$310$		0			0
$311$	−10.5182			−0.596435			−0.298217	−	0.954498i	$-0.596392\pi$
$311$	−10.5182			−0.596435			−0.298217	+	0.954498i	$0.596392\pi$
$312$		0			0
$313$	−13.7863			−0.779248			−0.389624	−	0.920974i	$-0.627395\pi$
$313$	−13.7863			−0.779248			−0.389624	+	0.920974i	$0.627395\pi$
$314$		0			0
$315$	−3.19289			−0.179899
$316$		0			0
$317$			8.18912i			0.459947i	0.973197	+	0.229973i	$0.0738639\pi$
$317$			8.18912i			0.459947i	−0.973197	+	0.229973i	$0.926136\pi$
$318$		0			0
$319$		−	11.6557i		−	0.652595i
$320$		0			0
$321$	4.49495			0.250884
$322$		0			0
$323$			0.199152i			0.0110811i
$324$		0			0
$325$	8.13718	−	16.8691i	0.451370	−	0.935728i
$326$		0			0
$327$		−	17.3106i		−	0.957276i
$328$		0			0
$329$	−4.98141			−0.274634
$330$		0			0
$331$			3.79915i			0.208820i	0.994534	+	0.104410i	$0.0332954\pi$
$331$			3.79915i			0.208820i	−0.994534	+	0.104410i	$0.966705\pi$
$332$		0			0
$333$		−	7.42708i		−	0.407002i
$334$		0			0
$335$	−42.4991			−2.32197
$336$		0			0
$337$	3.55322			0.193556			0.0967780	−	0.995306i	$-0.469146\pi$
$337$	3.55322			0.193556			0.0967780	+	0.995306i	$0.469146\pi$
$338$		0			0
$339$	−1.25912			−0.0683859
$340$		0			0
$341$	3.75386			0.203283
$342$		0			0
$343$		−	1.00000i		−	0.0539949i
$344$		0			0
$345$			13.3926i			0.721035i
$346$		0			0
$347$	−22.5491			−1.21050			−0.605250	−	0.796036i	$-0.706927\pi$
$347$	−22.5491			−1.21050			−0.605250	+	0.796036i	$0.706927\pi$
$348$		0			0
$349$			18.6520i			0.998416i	0.866482	+	0.499208i	$0.166376\pi$
$349$			18.6520i			0.998416i	−0.866482	+	0.499208i	$0.833624\pi$
$350$		0			0
$351$	−1.56649	+	3.24748i	−0.0836132	+	0.173337i
$352$		0			0
$353$			6.71955i			0.357645i	0.983881	+	0.178823i	$0.0572288\pi$
$353$			6.71955i			0.357645i	−0.983881	+	0.178823i	$0.942771\pi$
$354$		0			0
$355$	41.7207			2.21430
$356$		0			0
$357$			0.320653i			0.0169707i
$358$		0			0
$359$		−	21.0837i		−	1.11276i	−0.830929	−	0.556378i	$-0.812190\pi$
$359$		−	21.0837i		−	1.11276i	0.830929	−	0.556378i	$-0.187810\pi$
$360$		0			0
$361$	18.6143			0.979698
$362$		0			0
$363$	−14.6720			−0.770080
$364$		0			0
$365$	45.5398			2.38366
$366$		0			0
$367$	12.7822			0.667223			0.333612	−	0.942711i	$-0.391733\pi$
$367$	12.7822			0.667223			0.333612	+	0.942711i	$0.391733\pi$
$368$		0			0
$369$			9.49659i			0.494373i
$370$		0			0
$371$		−	2.94174i		−	0.152727i
$372$		0			0
$373$	−20.8134			−1.07768			−0.538838	−	0.842409i	$-0.681136\pi$
$373$	−20.8134			−1.07768			−0.538838	+	0.842409i	$0.681136\pi$
$374$		0			0
$375$		−	0.621083i		−	0.0320726i
$376$		0			0
$377$	3.60361	−	7.47059i	0.185595	−	0.384755i
$378$		0			0
$379$		−	7.34773i		−	0.377428i	−0.982032	−	0.188714i	$-0.939568\pi$
$379$		−	7.34773i		−	0.377428i	0.982032	−	0.188714i	$-0.0604318\pi$
$380$		0			0
$381$	−13.3212			−0.682464
$382$		0			0
$383$			10.9710i			0.560592i	0.959914	+	0.280296i	$0.0904326\pi$
$383$			10.9710i			0.560592i	−0.959914	+	0.280296i	$0.909567\pi$
$384$		0			0
$385$		−	16.1776i		−	0.824485i
$386$		0			0
$387$	11.0069			0.559510
$388$		0			0
$389$	37.9147			1.92235			0.961175	−	0.275938i	$-0.0889884\pi$
$389$	37.9147			1.92235			0.961175	+	0.275938i	$0.0889884\pi$
$390$		0			0
$391$	1.34498			0.0680188
$392$		0			0
$393$	16.3890			0.826718
$394$		0			0
$395$		−	11.3555i		−	0.571355i
$396$		0			0
$397$			31.7147i			1.59171i	0.605485	+	0.795857i	$0.292979\pi$
$397$			31.7147i			1.59171i	−0.605485	+	0.795857i	$0.707021\pi$
$398$		0			0
$399$	−0.621083			−0.0310930
$400$		0			0
$401$			9.42085i			0.470455i	0.971940	+	0.235227i	$0.0755835\pi$
$401$			9.42085i			0.470455i	−0.971940	+	0.235227i	$0.924417\pi$
$402$		0			0
$403$	2.40599	+	1.16059i	0.119851	+	0.0578129i
$404$		0			0
$405$			3.19289i			0.158656i
$406$		0			0
$407$	37.6312			1.86531
$408$		0			0
$409$		−	14.7779i		−	0.730719i	−0.930866	−	0.365360i	$-0.880946\pi$
$409$		−	14.7779i		−	0.730719i	0.930866	−	0.365360i	$-0.119054\pi$
$410$		0			0
$411$			5.32229i			0.262529i
$412$		0			0
$413$	0.680982			0.0335089
$414$		0			0
$415$	−29.7004			−1.45794
$416$		0			0
$417$	1.45364			0.0711850
$418$		0			0
$419$	−23.0375			−1.12546			−0.562728	−	0.826642i	$-0.690248\pi$
$419$	−23.0375			−1.12546			−0.562728	+	0.826642i	$0.690248\pi$
$420$		0			0
$421$		−	7.96983i		−	0.388426i	−0.980959	−	0.194213i	$-0.937785\pi$
$421$		−	7.96983i		−	0.388426i	0.980959	−	0.194213i	$-0.0622152\pi$
$422$		0			0
$423$			4.98141i			0.242205i
$424$		0			0
$425$	−1.66564			−0.0807953
$426$		0			0
$427$		−	3.13299i		−	0.151616i
$428$		0			0
$429$	−16.4542	−	7.93704i	−0.794414	−	0.383204i
$430$		0			0
$431$		−	25.9242i		−	1.24872i	−0.781135	−	0.624362i	$-0.785359\pi$
$431$		−	25.9242i		−	1.24872i	0.781135	−	0.624362i	$-0.214641\pi$
$432$		0			0
$433$	21.0826			1.01317			0.506583	−	0.862191i	$-0.330908\pi$
$433$	21.0826			1.01317			0.506583	+	0.862191i	$0.330908\pi$
$434$		0			0
$435$		−	7.34501i		−	0.352166i
$436$		0			0
$437$			2.60515i			0.124621i
$438$		0			0
$439$	−26.3729			−1.25871			−0.629356	−	0.777117i	$-0.716681\pi$
$439$	−26.3729			−1.25871			−0.629356	+	0.777117i	$0.716681\pi$
$440$		0			0
$441$	−1.00000			−0.0476190
$442$		0			0
$443$	29.9861			1.42468			0.712342	−	0.701832i	$-0.247634\pi$
$443$	29.9861			1.42468			0.712342	+	0.701832i	$0.247634\pi$
$444$		0			0
$445$	43.5735			2.06558
$446$		0			0
$447$			12.0906i			0.571864i
$448$		0			0
$449$		−	10.4829i		−	0.494717i	−0.968924	−	0.247359i	$-0.920437\pi$
$449$		−	10.4829i		−	0.494717i	0.968924	−	0.247359i	$-0.0795626\pi$
$450$		0			0
$451$	−48.1169			−2.26573
$452$		0			0
$453$			9.62467i			0.452206i
$454$		0			0
$455$	5.00163	−	10.3688i	0.234480	−	0.486098i
$456$		0			0
$457$			13.5326i			0.633031i	0.948587	+	0.316515i	$0.102513\pi$
$457$			13.5326i			0.633031i	−0.948587	+	0.316515i	$0.897487\pi$
$458$		0			0
$459$	0.320653			0.0149668
$460$		0			0
$461$			11.2990i			0.526247i	0.964762	+	0.263123i	$0.0847527\pi$
$461$			11.2990i			0.526247i	−0.964762	+	0.263123i	$0.915247\pi$
$462$		0			0
$463$			21.2399i			0.987104i	0.869716	+	0.493552i	$0.164302\pi$
$463$			21.2399i			0.987104i	−0.869716	+	0.493552i	$0.835698\pi$
$464$		0			0
$465$	2.36555			0.109700
$466$		0			0
$467$	26.9220			1.24580			0.622902	−	0.782300i	$-0.285954\pi$
$467$	26.9220			1.24580			0.622902	+	0.782300i	$0.285954\pi$
$468$		0			0
$469$	−13.3106			−0.614624
$470$		0			0
$471$	−14.5495			−0.670403
$472$		0			0
$473$			55.7690i			2.56426i
$474$		0			0
$475$		−	3.22623i		−	0.148030i
$476$		0			0
$477$	−2.94174			−0.134693
$478$		0			0
$479$			7.79013i			0.355940i	0.984036	+	0.177970i	$0.0569530\pi$
$479$			7.79013i			0.355940i	−0.984036	+	0.177970i	$0.943047\pi$
$480$		0			0
$481$	24.1193	+	11.6345i	1.09974	+	0.530487i
$482$		0			0
$483$			4.19452i			0.190857i
$484$		0			0
$485$	15.1032			0.685800
$486$		0			0
$487$		−	8.45796i		−	0.383267i	−0.981467	−	0.191633i	$-0.938622\pi$
$487$		−	8.45796i		−	0.383267i	0.981467	−	0.191633i	$-0.0613784\pi$
$488$		0			0
$489$			11.0446i			0.499453i
$490$		0			0
$491$	28.8713			1.30294			0.651472	−	0.758672i	$-0.274152\pi$
$491$	28.8713			1.30294			0.651472	+	0.758672i	$0.274152\pi$
$492$		0			0
$493$	−0.737639			−0.0332216
$494$		0			0
$495$	−16.1776			−0.727128
$496$		0			0
$497$	13.0668			0.586124
$498$		0			0
$499$			24.5433i			1.09871i	0.835589	+	0.549355i	$0.185127\pi$
$499$			24.5433i			1.09871i	−0.835589	+	0.549355i	$0.814873\pi$
$500$		0			0
$501$			15.4181i			0.688829i
$502$		0			0
$503$	36.9083			1.64566			0.822830	−	0.568287i	$-0.192394\pi$
$503$	36.9083			1.64566			0.822830	+	0.568287i	$0.192394\pi$
$504$		0			0
$505$			8.47884i			0.377303i
$506$		0			0
$507$	−8.09220	−	10.1743i	−0.359387	−	0.451856i
$508$		0			0
$509$			21.8140i			0.966888i	0.875375	+	0.483444i	$0.160614\pi$
$509$			21.8140i			0.966888i	−0.875375	+	0.483444i	$0.839386\pi$
$510$		0			0
$511$	14.2629			0.630954
$512$		0			0
$513$			0.621083i			0.0274215i
$514$		0			0
$515$		−	38.2403i		−	1.68507i
$516$		0			0
$517$	−25.2396			−1.11004
$518$		0			0
$519$	−3.29685			−0.144715
$520$		0			0
$521$	−23.3390			−1.02250			−0.511249	−	0.859432i	$-0.670817\pi$
$521$	−23.3390			−1.02250			−0.511249	+	0.859432i	$0.670817\pi$
$522$		0			0
$523$	−23.3244			−1.01991			−0.509953	−	0.860202i	$-0.670337\pi$
$523$	−23.3244			−1.01991			−0.509953	+	0.860202i	$0.670337\pi$
$524$		0			0
$525$		−	5.19452i		−	0.226707i
$526$		0			0
$527$		−	0.237566i		−	0.0103485i
$528$		0			0
$529$	−5.40599			−0.235043
$530$		0			0
$531$		−	0.680982i		−	0.0295521i
$532$		0			0
$533$	−30.8399	−	14.8763i	−1.33583	−	0.644366i
$534$		0			0
$535$			14.3519i			0.620485i
$536$		0			0
$537$	5.48810			0.236829
$538$		0			0
$539$		−	5.06675i		−	0.218241i
$540$		0			0
$541$			33.1092i			1.42347i	0.702446	+	0.711737i	$0.252091\pi$
$541$			33.1092i			1.42347i	−0.702446	+	0.711737i	$0.747909\pi$
$542$		0			0
$543$	−9.30729			−0.399414
$544$		0			0
$545$	55.2706			2.36753
$546$		0			0
$547$	35.5738			1.52103			0.760513	−	0.649323i	$-0.224947\pi$
$547$	35.5738			1.52103			0.760513	+	0.649323i	$0.224947\pi$
$548$		0			0
$549$	−3.13299			−0.133713
$550$		0			0
$551$		−	1.42876i		−	0.0608672i
$552$		0			0
$553$		−	3.55649i		−	0.151237i
$554$		0			0
$555$	23.7138			1.00660
$556$		0			0
$557$		−	4.73187i		−	0.200496i	−0.994962	−	0.100248i	$-0.968036\pi$
$557$		−	4.73187i		−	0.200496i	0.994962	−	0.100248i	$-0.0319636\pi$
$558$		0			0
$559$	−17.2422	+	35.7445i	−0.729266	+	1.51183i
$560$		0			0
$561$			1.62467i			0.0685936i
$562$		0			0
$563$	29.9671			1.26296			0.631482	−	0.775390i	$-0.282447\pi$
$563$	29.9671			1.26296			0.631482	+	0.775390i	$0.282447\pi$
$564$		0			0
$565$		−	4.02022i		−	0.169132i
$566$		0			0
$567$			1.00000i			0.0419961i
$568$		0			0
$569$	2.39228			0.100290			0.0501449	−	0.998742i	$-0.484032\pi$
$569$	2.39228			0.100290			0.0501449	+	0.998742i	$0.484032\pi$
$570$		0			0
$571$	14.1923			0.593929			0.296965	−	0.954889i	$-0.404026\pi$
$571$	14.1923			0.593929			0.296965	+	0.954889i	$0.404026\pi$
$572$		0			0
$573$	13.0656			0.545825
$574$		0			0
$575$	−21.7885			−0.908644
$576$		0			0
$577$		−	15.8547i		−	0.660039i	−0.943974	−	0.330020i	$-0.892945\pi$
$577$		−	15.8547i		−	0.660039i	0.943974	−	0.330020i	$-0.107055\pi$
$578$		0			0
$579$		−	15.3757i		−	0.638991i
$580$		0			0
$581$	−9.30207			−0.385915
$582$		0			0
$583$		−	14.9051i		−	0.617304i
$584$		0			0
$585$	−10.3688	−	5.00163i	−0.428698	−	0.206792i
$586$		0			0
$587$			7.14509i			0.294910i	0.989069	+	0.147455i	$0.0471081\pi$
$587$			7.14509i			0.294910i	−0.989069	+	0.147455i	$0.952892\pi$
$588$		0			0
$589$	0.460149			0.0189601
$590$		0			0
$591$		−	10.8961i		−	0.448204i
$592$		0			0
$593$			20.9329i			0.859612i	0.902921	+	0.429806i	$0.141418\pi$
$593$			20.9329i			0.859612i	−0.902921	+	0.429806i	$0.858582\pi$
$594$		0			0
$595$	−1.02381			−0.0419720
$596$		0			0
$597$	17.5945			0.720095
$598$		0			0
$599$	7.75891			0.317020			0.158510	−	0.987357i	$-0.449331\pi$
$599$	7.75891			0.317020			0.158510	+	0.987357i	$0.449331\pi$
$600$		0			0
$601$	36.0927			1.47225			0.736126	−	0.676844i	$-0.236653\pi$
$601$	36.0927			1.47225			0.736126	+	0.676844i	$0.236653\pi$
$602$		0			0
$603$			13.3106i			0.542048i
$604$		0			0
$605$		−	46.8460i		−	1.90456i
$606$		0			0
$607$	−20.1135			−0.816381			−0.408191	−	0.912897i	$-0.633840\pi$
$607$	−20.1135			−0.816381			−0.408191	+	0.912897i	$0.633840\pi$
$608$		0			0
$609$		−	2.30043i		−	0.0932181i
$610$		0			0
$611$	−16.1770	−	7.80335i	−0.654452	−	0.315690i
$612$		0			0
$613$		−	33.2312i		−	1.34220i	−0.741369	−	0.671098i	$-0.765823\pi$
$613$		−	33.2312i		−	1.34220i	0.741369	−	0.671098i	$-0.234177\pi$
$614$		0			0
$615$	−30.3215			−1.22268
$616$		0			0
$617$			10.2263i			0.411695i	0.978584	+	0.205848i	$0.0659951\pi$
$617$			10.2263i			0.411695i	−0.978584	+	0.205848i	$0.934005\pi$
$618$		0			0
$619$		−	13.2526i		−	0.532667i	−0.963881	−	0.266334i	$-0.914188\pi$
$619$		−	13.2526i		−	0.532667i	0.963881	−	0.266334i	$-0.0858123\pi$
$620$		0			0
$621$	4.19452			0.168320
$622$		0			0
$623$	13.6470			0.546757
$624$		0			0
$625$	−23.9896			−0.959582
$626$		0			0
$627$	−3.14688			−0.125674
$628$		0			0
$629$		−	2.38151i		−	0.0949572i
$630$		0			0
$631$			18.7748i			0.747414i	0.927547	+	0.373707i	$0.121913\pi$
$631$			18.7748i			0.747414i	−0.927547	+	0.373707i	$0.878087\pi$
$632$		0			0
$633$	20.9354			0.832109
$634$		0			0
$635$		−	42.5330i		−	1.68787i
$636$		0			0
$637$	1.56649	−	3.24748i	0.0620667	−	0.128670i
$638$		0			0
$639$		−	13.0668i		−	0.516913i
$640$		0			0
$641$	−36.2096			−1.43019			−0.715097	−	0.699025i	$-0.753617\pi$
$641$	−36.2096			−1.43019			−0.715097	+	0.699025i	$0.753617\pi$
$642$		0			0
$643$		−	22.2725i		−	0.878342i	−0.898403	−	0.439171i	$-0.855272\pi$
$643$		−	22.2725i		−	0.878342i	0.898403	−	0.439171i	$-0.144728\pi$
$644$		0			0
$645$			35.1436i			1.38378i
$646$		0			0
$647$	15.1880			0.597102			0.298551	−	0.954394i	$-0.403497\pi$
$647$	15.1880			0.597102			0.298551	+	0.954394i	$0.403497\pi$
$648$		0			0
$649$	3.45037			0.135439
$650$		0			0
$651$	0.740881			0.0290374
$652$		0			0
$653$	−10.7009			−0.418760			−0.209380	−	0.977834i	$-0.567145\pi$
$653$	−10.7009			−0.418760			−0.209380	+	0.977834i	$0.567145\pi$
$654$		0			0
$655$			52.3283i			2.04464i
$656$		0			0
$657$		−	14.2629i		−	0.556449i
$658$		0			0
$659$	21.6036			0.841558			0.420779	−	0.907163i	$-0.361757\pi$
$659$	21.6036			0.841558			0.420779	+	0.907163i	$0.361757\pi$
$660$		0			0
$661$			20.0583i			0.780176i	0.920778	+	0.390088i	$0.127555\pi$
$661$			20.0583i			0.780176i	−0.920778	+	0.390088i	$0.872445\pi$
$662$		0			0
$663$	−0.502300	+	1.04131i	−0.0195077	+	0.0404412i
$664$		0			0
$665$		−	1.98305i		−	0.0768993i
$666$		0			0
$667$	−9.64920			−0.373619
$668$		0			0
$669$		−	19.1267i		−	0.739479i
$670$		0			0
$671$		−	15.8741i		−	0.612812i
$672$		0			0
$673$	−24.1664			−0.931546			−0.465773	−	0.884904i	$-0.654224\pi$
$673$	−24.1664			−0.931546			−0.465773	+	0.884904i	$0.654224\pi$
$674$		0			0
$675$	−5.19452			−0.199937
$676$		0			0
$677$	11.8161			0.454131			0.227065	−	0.973880i	$-0.427087\pi$
$677$	11.8161			0.454131			0.227065	+	0.973880i	$0.427087\pi$
$678$		0			0
$679$	4.73026			0.181531
$680$		0			0
$681$		−	29.7218i		−	1.13894i
$682$		0			0
$683$			10.7127i			0.409909i	0.978772	+	0.204954i	$0.0657046\pi$
$683$			10.7127i			0.409909i	−0.978772	+	0.204954i	$0.934295\pi$
$684$		0			0
$685$	−16.9935			−0.649287
$686$		0			0
$687$			9.99265i			0.381243i
$688$		0			0
$689$	4.60821	−	9.55322i	0.175559	−	0.363949i
$690$		0			0
$691$		−	6.89360i		−	0.262245i	−0.991366	−	0.131122i	$-0.958142\pi$
$691$		−	6.89360i		−	0.262245i	0.991366	−	0.131122i	$-0.0418581\pi$
$692$		0			0
$693$	−5.06675			−0.192470
$694$		0			0
$695$			4.64131i			0.176055i
$696$		0			0
$697$			3.04511i			0.115342i
$698$		0			0
$699$	24.9253			0.942760
$700$		0			0
$701$	−27.4337			−1.03615			−0.518077	−	0.855334i	$-0.673352\pi$
$701$	−27.4337			−1.03615			−0.518077	+	0.855334i	$0.673352\pi$
$702$		0			0
$703$	4.61284			0.173976
$704$		0			0
$705$	−15.9051			−0.599020
$706$		0			0
$707$			2.65554i			0.0998719i
$708$		0			0
$709$			31.4271i			1.18027i	0.807305	+	0.590134i	$0.200925\pi$
$709$			31.4271i			1.18027i	−0.807305	+	0.590134i	$0.799075\pi$
$710$		0			0
$711$	−3.55649			−0.133379
$712$		0			0
$713$		−	3.10764i		−	0.116382i
$714$		0			0
$715$	25.3421	−	52.5363i	0.947739	−	1.96474i
$716$		0			0
$717$		−	9.66762i		−	0.361044i
$718$		0			0
$719$	−5.26994			−0.196536			−0.0982679	−	0.995160i	$-0.531330\pi$
$719$	−5.26994			−0.196536			−0.0982679	+	0.995160i	$0.531330\pi$
$720$		0			0
$721$		−	11.9767i		−	0.446036i
$722$		0			0
$723$			23.2867i			0.866042i
$724$		0			0
$725$	11.9496			0.443798
$726$		0			0
$727$	−14.2430			−0.528245			−0.264122	−	0.964489i	$-0.585082\pi$
$727$	−14.2430			−0.528245			−0.264122	+	0.964489i	$0.585082\pi$
$728$		0			0
$729$	1.00000			0.0370370
$730$		0			0
$731$	3.52938			0.130539
$732$		0			0
$733$			48.0244i			1.77382i	0.461942	+	0.886910i	$0.347153\pi$
$733$			48.0244i			1.77382i	−0.461942	+	0.886910i	$0.652847\pi$
$734$		0			0
$735$		−	3.19289i		−	0.117771i
$736$		0			0
$737$	−67.4413			−2.48423
$738$		0			0
$739$		−	35.0593i		−	1.28968i	−0.764318	−	0.644839i	$-0.776924\pi$
$739$		−	35.0593i		−	1.28968i	0.764318	−	0.644839i	$-0.223076\pi$
$740$		0			0
$741$	−2.01695	−	0.972923i	−0.0740946	−	0.0357412i
$742$		0			0
$743$			9.82132i			0.360309i	0.983638	+	0.180155i	$0.0576598\pi$
$743$			9.82132i			0.360309i	−0.983638	+	0.180155i	$0.942340\pi$
$744$		0			0
$745$	−38.6038			−1.41433
$746$		0			0
$747$			9.30207i			0.340345i
$748$		0			0
$749$			4.49495i			0.164242i
$750$		0			0
$751$	21.3356			0.778548			0.389274	−	0.921122i	$-0.372726\pi$
$751$	21.3356			0.778548			0.389274	+	0.921122i	$0.372726\pi$
$752$		0			0
$753$	−4.98990			−0.181842
$754$		0			0
$755$	−30.7305			−1.11840
$756$		0			0
$757$	34.6312			1.25869			0.629346	−	0.777125i	$-0.283323\pi$
$757$	34.6312			1.25869			0.629346	+	0.777125i	$0.283323\pi$
$758$		0			0
$759$			21.2526i			0.771421i
$760$		0			0
$761$			23.3598i			0.846793i	0.905944	+	0.423397i	$0.139162\pi$
$761$			23.3598i			0.846793i	−0.905944	+	0.423397i	$0.860838\pi$
$762$		0			0
$763$	17.3106			0.626684
$764$		0			0
$765$			1.02381i			0.0370158i
$766$		0			0
$767$	2.21147	+	1.06675i	0.0798517	+	0.0385183i
$768$		0			0
$769$		−	43.5942i		−	1.57205i	−0.618197	−	0.786023i	$-0.712137\pi$
$769$		−	43.5942i		−	1.57205i	0.618197	−	0.786023i	$-0.287863\pi$
$770$		0			0
$771$	−18.7923			−0.676789
$772$		0			0
$773$		−	21.6674i		−	0.779324i	−0.920958	−	0.389662i	$-0.872592\pi$
$773$		−	21.6674i		−	0.779324i	0.920958	−	0.389662i	$-0.127408\pi$
$774$		0			0
$775$			3.84852i			0.138243i
$776$		0			0
$777$	7.42708			0.266445
$778$		0			0
$779$	−5.89817			−0.211324
$780$		0			0
$781$	66.2060			2.36904
$782$		0			0
$783$	−2.30043			−0.0822107
$784$		0			0
$785$		−	46.4547i		−	1.65804i
$786$		0			0
$787$		−	15.6586i		−	0.558169i	−0.960267	−	0.279084i	$-0.909969\pi$
$787$		−	15.6586i		−	0.558169i	0.960267	−	0.279084i	$-0.0900309\pi$
$788$		0			0
$789$	−10.3110			−0.367083
$790$		0			0
$791$		−	1.25912i		−	0.0447691i
$792$		0			0
$793$	4.90780	−	10.1743i	0.174281	−	0.361300i
$794$		0			0
$795$		−	9.39263i		−	0.333122i
$796$		0			0
$797$	−1.83632			−0.0650458			−0.0325229	−	0.999471i	$-0.510354\pi$
$797$	−1.83632			−0.0650458			−0.0325229	+	0.999471i	$0.510354\pi$
$798$		0			0
$799$			1.59730i			0.0565085i
$800$		0			0
$801$		−	13.6470i		−	0.482195i
$802$		0			0
$803$	72.2667			2.55024
$804$		0			0
$805$	−13.3926			−0.472028
$806$		0			0
$807$	10.0821			0.354907
$808$		0			0
$809$	2.51598			0.0884572			0.0442286	−	0.999021i	$-0.485917\pi$
$809$	2.51598			0.0884572			0.0442286	+	0.999021i	$0.485917\pi$
$810$		0			0
$811$		−	29.2083i		−	1.02564i	−0.858496	−	0.512820i	$-0.828601\pi$
$811$		−	29.2083i		−	1.02564i	0.858496	−	0.512820i	$-0.171399\pi$
$812$		0			0
$813$			20.9105i			0.733365i
$814$		0			0
$815$	−35.2641			−1.23525
$816$		0			0
$817$			6.83617i			0.239167i
$818$		0			0
$819$	−3.24748	−	1.56649i	−0.113476	−	0.0547377i
$820$		0			0
$821$		−	26.6662i		−	0.930658i	−0.885138	−	0.465329i	$-0.845936\pi$
$821$		−	26.6662i		−	0.930658i	0.885138	−	0.465329i	$-0.154064\pi$
$822$		0			0
$823$	30.4174			1.06029			0.530143	−	0.847908i	$-0.322138\pi$
$823$	30.4174			1.06029			0.530143	+	0.847908i	$0.322138\pi$
$824$		0			0
$825$		−	26.3194i		−	0.916323i
$826$		0			0
$827$		−	24.9404i		−	0.867263i	−0.901090	−	0.433632i	$-0.857232\pi$
$827$		−	24.9404i		−	0.867263i	0.901090	−	0.433632i	$-0.142768\pi$
$828$		0			0
$829$	−41.4953			−1.44119			−0.720596	−	0.693356i	$-0.756132\pi$
$829$	−41.4953			−1.44119			−0.720596	+	0.693356i	$0.756132\pi$
$830$		0			0
$831$	11.8610			0.411454
$832$		0			0
$833$	−0.320653			−0.0111100
$834$		0			0
$835$	−49.2281			−1.70361
$836$		0			0
$837$		−	0.740881i		−	0.0256086i
$838$		0			0
$839$			52.2236i			1.80296i	0.432822	+	0.901480i	$0.357518\pi$
$839$			52.2236i			1.80296i	−0.432822	+	0.901480i	$0.642482\pi$
$840$		0			0
$841$	−23.7080			−0.817518
$842$		0			0
$843$			18.0534i			0.621792i
$844$		0			0
$845$	32.4854	−	25.8375i	1.11753	−	0.888836i
$846$		0			0
$847$		−	14.6720i		−	0.504136i
$848$		0			0
$849$	−6.50557			−0.223271
$850$		0			0
$851$		−	31.1531i		−	1.06791i
$852$		0			0
$853$		−	36.1229i		−	1.23683i	−0.785854	−	0.618413i	$-0.787776\pi$
$853$		−	36.1229i		−	1.23683i	0.785854	−	0.618413i	$-0.212224\pi$
$854$		0			0
$855$	−1.98305			−0.0678188
$856$		0			0
$857$	56.3289			1.92416			0.962079	−	0.272772i	$-0.0879405\pi$
$857$	56.3289			1.92416			0.962079	+	0.272772i	$0.0879405\pi$
$858$		0			0
$859$	40.5467			1.38344			0.691718	−	0.722168i	$-0.256854\pi$
$859$	40.5467			1.38344			0.691718	+	0.722168i	$0.256854\pi$
$860$		0			0
$861$	−9.49659			−0.323643
$862$		0			0
$863$		−	37.6243i		−	1.28074i	−0.768065	−	0.640372i	$-0.778780\pi$
$863$		−	37.6243i		−	1.28074i	0.768065	−	0.640372i	$-0.221220\pi$
$864$		0			0
$865$		−	10.5265i		−	0.357910i
$866$		0			0
$867$	−16.8972			−0.573858
$868$		0			0
$869$		−	18.0198i		−	0.611281i
$870$		0			0
$871$	−43.2257	−	20.8509i	−1.46465	−	0.706506i
$872$		0			0
$873$		−	4.73026i		−	0.160095i
$874$		0			0
$875$	0.621083			0.0209964
$876$		0			0
$877$		−	55.4441i		−	1.87221i	−0.351717	−	0.936107i	$-0.614402\pi$
$877$		−	55.4441i		−	1.87221i	0.351717	−	0.936107i	$-0.385598\pi$
$878$		0			0
$879$		−	22.7392i		−	0.766976i
$880$		0			0
$881$	30.3491			1.02249			0.511244	−	0.859436i	$-0.329185\pi$
$881$	30.3491			1.02249			0.511244	+	0.859436i	$0.329185\pi$
$882$		0			0
$883$	15.2108			0.511883			0.255942	−	0.966692i	$-0.417615\pi$
$883$	15.2108			0.511883			0.255942	+	0.966692i	$0.417615\pi$
$884$		0			0
$885$	2.17430			0.0730883
$886$		0			0
$887$	35.1126			1.17897			0.589483	−	0.807781i	$-0.299331\pi$
$887$	35.1126			1.17897			0.589483	+	0.807781i	$0.299331\pi$
$888$		0			0
$889$		−	13.3212i		−	0.446778i
$890$		0			0
$891$			5.06675i			0.169743i
$892$		0			0
$893$	−3.09387			−0.103532
$894$		0			0
$895$			17.5229i			0.585725i
$896$		0			0
$897$	−6.57069	+	13.6216i	−0.219389	+	0.454812i
$898$		0			0
$899$			1.70435i			0.0568431i
$900$		0			0
$901$	−0.943275			−0.0314251
$902$		0			0
$903$			11.0069i			0.366285i
$904$		0			0
$905$		−	29.7171i		−	0.987830i
$906$		0			0
$907$	−35.8362			−1.18992			−0.594960	−	0.803755i	$-0.702832\pi$
$907$	−35.8362			−1.18992			−0.594960	+	0.803755i	$0.702832\pi$
$908$		0			0
$909$	2.65554			0.0880787
$910$		0			0
$911$	−22.1201			−0.732872			−0.366436	−	0.930443i	$-0.619422\pi$
$911$	−22.1201			−0.732872			−0.366436	+	0.930443i	$0.619422\pi$
$912$		0			0
$913$	−47.1313			−1.55982
$914$		0			0
$915$		−	10.0033i		−	0.330698i
$916$		0			0
$917$			16.3890i			0.541214i
$918$		0			0
$919$	−2.38612			−0.0787108			−0.0393554	−	0.999225i	$-0.512530\pi$
$919$	−2.38612			−0.0787108			−0.0393554	+	0.999225i	$0.512530\pi$
$920$		0			0
$921$			31.3959i			1.03453i
$922$		0			0
$923$	42.4340	+	20.4690i	1.39673	+	0.673745i
$924$		0			0
$925$			38.5801i			1.26851i
$926$		0			0
$927$	−11.9767			−0.393367
$928$		0			0
$929$			31.9585i			1.04852i	0.851557	+	0.524262i	$0.175659\pi$
$929$			31.9585i			1.04852i	−0.851557	+	0.524262i	$0.824341\pi$
$930$		0			0
$931$		−	0.621083i		−	0.0203552i
$932$		0			0
$933$	−10.5182			−0.344352
$934$		0			0
$935$	−5.18738			−0.169645
$936$		0			0
$937$	36.1613			1.18134			0.590669	−	0.806914i	$-0.298864\pi$
$937$	36.1613			1.18134			0.590669	+	0.806914i	$0.298864\pi$
$938$		0			0
$939$	−13.7863			−0.449899
$940$		0			0
$941$			5.17695i			0.168764i	0.996433	+	0.0843819i	$0.0268916\pi$
$941$			5.17695i			0.168764i	−0.996433	+	0.0843819i	$0.973108\pi$
$942$		0			0
$943$			39.8336i			1.29716i
$944$		0			0
$945$	−3.19289			−0.103865
$946$		0			0
$947$		−	48.4868i		−	1.57561i	−0.615926	−	0.787804i	$-0.711218\pi$
$947$		−	48.4868i		−	1.57561i	0.615926	−	0.787804i	$-0.288782\pi$
$948$		0			0
$949$	46.3185	+	22.3428i	1.50356	+	0.725277i
$950$		0			0
$951$			8.18912i			0.265550i
$952$		0			0
$953$	1.29413			0.0419209			0.0209604	−	0.999780i	$-0.493328\pi$
$953$	1.29413			0.0419209			0.0209604	+	0.999780i	$0.493328\pi$
$954$		0			0
$955$			41.7171i			1.34993i
$956$		0			0
$957$		−	11.6557i		−	0.376776i
$958$		0			0
$959$	−5.32229			−0.171866
$960$		0			0
$961$	30.4511			0.982293
$962$		0			0
$963$	4.49495			0.144848
$964$		0			0
$965$	49.0928			1.58035
$966$		0			0
$967$		−	20.3179i		−	0.653380i	−0.945132	−	0.326690i	$-0.894067\pi$
$967$		−	20.3179i		−	0.653380i	0.945132	−	0.326690i	$-0.105933\pi$
$968$		0			0
$969$			0.199152i			0.00639768i
$970$		0			0
$971$	−1.77614			−0.0569992			−0.0284996	−	0.999594i	$-0.509073\pi$
$971$	−1.77614			−0.0569992			−0.0284996	+	0.999594i	$0.509073\pi$
$972$		0			0
$973$			1.45364i			0.0466015i
$974$		0			0
$975$	8.13718	−	16.8691i	0.260598	−	0.540243i
$976$		0			0
$977$		−	52.9458i		−	1.69389i	−0.531684	−	0.846943i	$-0.678441\pi$
$977$		−	52.9458i		−	1.69389i	0.531684	−	0.846943i	$-0.321559\pi$
$978$		0			0
$979$	69.1462			2.20992
$980$		0			0
$981$		−	17.3106i		−	0.552683i
$982$		0			0
$983$			41.4838i			1.32313i	0.749889	+	0.661563i	$0.230107\pi$
$983$			41.4838i			1.32313i	−0.749889	+	0.661563i	$0.769893\pi$
$984$		0			0
$985$	34.7899			1.10850
$986$		0			0
$987$	−4.98141			−0.158560
$988$		0			0
$989$	46.1685			1.46807
$990$		0			0
$991$	−49.8704			−1.58419			−0.792093	−	0.610400i	$-0.791009\pi$
$991$	−49.8704			−1.58419			−0.792093	+	0.610400i	$0.791009\pi$
$992$		0			0
$993$			3.79915i			0.120562i
$994$		0			0
$995$			56.1772i			1.78094i
$996$		0			0
$997$	−12.0312			−0.381032			−0.190516	−	0.981684i	$-0.561016\pi$
$997$	−12.0312			−0.381032			−0.190516	+	0.981684i	$0.561016\pi$
$998$		0			0
$999$		−	7.42708i		−	0.234983i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4368.2.h.t.337.9		10
4.3	odd	2		2184.2.h.e.337.9	yes	10
13.12	even	2	inner	4368.2.h.t.337.2		10
52.51	odd	2		2184.2.h.e.337.2	✓	10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2184.2.h.e.337.2	✓	10	52.51	odd	2
2184.2.h.e.337.9	yes	10	4.3	odd	2
4368.2.h.t.337.2		10	13.12	even	2	inner
4368.2.h.t.337.9		10	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 4368 = 2^{4} \cdot 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4368.h (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+1.00000 q^{3} +3.19289i q^{5} +1.00000i q^{7} +1.00000 q^{9} +O(q^{10})\)	\(q+1.00000 q^{3} +3.19289i q^{5} +1.00000i q^{7} +1.00000 q^{9} +5.06675i q^{11} +(-1.56649 + 3.24748i) q^{13} +3.19289i q^{15} +0.320653 q^{17} +0.621083i q^{19} +1.00000i q^{21} +4.19452 q^{23} -5.19452 q^{25} +1.00000 q^{27} -2.30043 q^{29} -0.740881i q^{31} +5.06675i q^{33} -3.19289 q^{35} -7.42708i q^{37} +(-1.56649 + 3.24748i) q^{39} +9.49659i q^{41} +11.0069 q^{43} +3.19289i q^{45} +4.98141i q^{47} -1.00000 q^{49} +0.320653 q^{51} -2.94174 q^{53} -16.1776 q^{55} +0.621083i q^{57} -0.680982i q^{59} -3.13299 q^{61} +1.00000i q^{63} +(-10.3688 - 5.00163i) q^{65} +13.3106i q^{67} +4.19452 q^{69} -13.0668i q^{71} -14.2629i q^{73} -5.19452 q^{75} -5.06675 q^{77} -3.55649 q^{79} +1.00000 q^{81} +9.30207i q^{83} +1.02381i q^{85} -2.30043 q^{87} -13.6470i q^{89} +(-3.24748 - 1.56649i) q^{91} -0.740881i q^{93} -1.98305 q^{95} -4.73026i q^{97} +5.06675i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 10 q^{3} + 10 q^{9}+O(q^{10}) \) 10 * q + 10 * q^3 + 10 * q^9	\( 10 q + 10 q^{3} + 10 q^{9} + 2 q^{13} - 2 q^{23} - 8 q^{25} + 10 q^{27} - 2 q^{29} - 6 q^{35} + 2 q^{39} + 34 q^{43} - 10 q^{49} - 2 q^{53} - 64 q^{55} + 4 q^{61} + 2 q^{65} - 2 q^{69} - 8 q^{75} + 16 q^{77} + 38 q^{79} + 10 q^{81} - 2 q^{87} + 34 q^{95}+O(q^{100}) \) 10 * q + 10 * q^3 + 10 * q^9 + 2 * q^13 - 2 * q^23 - 8 * q^25 + 10 * q^27 - 2 * q^29 - 6 * q^35 + 2 * q^39 + 34 * q^43 - 10 * q^49 - 2 * q^53 - 64 * q^55 + 4 * q^61 + 2 * q^65 - 2 * q^69 - 8 * q^75 + 16 * q^77 + 38 * q^79 + 10 * q^81 - 2 * q^87 + 34 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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